Quasitriangular structures on pointed Hopf algebras of rank one (Q2826320)

In the paper [J. Algebra 302, No. 1, 214--230 (2006; Zbl 1126.16028)], \textit{L. Krop} and \textit{D. E. Radford} gave the classification of finite dimensional pointed Hopf algebras of rank one over an algebraically closed field of characteristic zero in terms of certain \textit{group datum} \({\mathbf D} = (G, \chi, g, \mu)\), where \(G\) is a finite group, \(\chi\) is a character on \(G\), \(g \in G\) is a central element, and \(\mu\) is a scalar satisfying appropriate conditions.NEWLINENEWLINEThe main result of the paper under review is a necessary and sufficient condition for a finite dimensional pointed Hopf algebra of rank one to be quasitriangular. As it turns out, the existence of quasitriangular structures on the Hopf algebra \(H_{\mathbf D}\) corresponding to \({\mathbf D}\) is strongly related to the order of the root of unity \(\chi(g)\). In particular, it is shown that \(H_{\mathbf D}\) is a group algebra if \(\chi(g) = 1\), and it admits no quasitriangular structure if the order of \(\chi(g)\) is bigger than \(2\).NEWLINENEWLINEAs an application, the author determines all quasitriangular structures on a finite dimensional pointed Hopf algebra of rank one with cyclic group of group-like elements.